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A328407 G.f. A(x) satisfies: A(x) = A(x^2) + x * (1 + x) / (1 - x)^3. 1
1, 5, 9, 21, 25, 45, 49, 85, 81, 125, 121, 189, 169, 245, 225, 341, 289, 405, 361, 525, 441, 605, 529, 765, 625, 845, 729, 1029, 841, 1125, 961, 1365, 1089, 1445, 1225, 1701, 1369, 1805, 1521, 2125, 1681, 2205, 1849, 2541, 2025, 2645, 2209, 3069, 2401, 3125, 2601, 3549, 2809, 3645, 3025 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..55.

FORMULA

G.f.: Sum_{k>=0} x^(2^k) * (1 + x^(2^k)) / (1 - x^(2^k))^3.

G.f.: (1/3) * Sum_{k>=1} J_2(2*k) * x^k / (1 - x^k), where J_2() is the Jordan function (A007434).

Dirichlet g.f.: zeta(s-2) / (1 - 2^(-s)).

a(2*n) = a(n) + 4*n^2, a(2*n+1) = (2*n + 1)^2.

a(n) = Sum_{d|n} A209229(n/d) * d^2.

Product_{n>=1} (1 + x^n)^a(n) = g.f. for A023871.

Sum_{k=1..n} a(k) ~ 8*n^3/21. - Vaclav Kotesovec, Oct 15 2019

MATHEMATICA

nmax = 55; CoefficientList[Series[Sum[x^(2^k) (1 + x^(2^k))/(1 - x^(2^k))^3, {k, 0, Floor[Log[2, nmax]] + 1}], {x, 0, nmax}], x] // Rest

a[n_] := If[EvenQ[n], a[n/2] + n^2, n^2]; Table[a[n], {n, 1, 55}]

Table[DivisorSum[n, Boole[IntegerQ[Log[2, n/#]]] #^2 &], {n, 1, 55}]

PROG

(MAGMA) [n eq 1 select 1 else IsOdd(n) select n^2 else Self(n div 2)+n^2 :n in [1..55]]; // Marius A. Burtea, Oct 15 2019

CROSSREFS

Cf. A000290, A001511, A007434, A016754, A023871, A129527, A209229, A328408.

Sequence in context: A246335 A255264 A321170 * A147407 A186297 A272986

Adjacent sequences:  A328404 A328405 A328406 * A328408 A328409 A328410

KEYWORD

nonn,mult

AUTHOR

Ilya Gutkovskiy, Oct 14 2019

STATUS

approved

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Last modified February 28 05:46 EST 2020. Contains 332321 sequences. (Running on oeis4.)